It is impossible to predict what may happen in a hypothetical future court proceeding to modify child custody and visitation. The best thing to do is closely document the problems as they arise. Ask the supervisor to prepare detailed reports any time something bad happens on his or her watch, and to provide you with a copy. If and when the father files a motion to modify custody, you should retain an experienced family law attorney in your area to ensure your rights are protected.
I am not your attorney. This comment does not create an attorney-client relationship. I have not considered the circumstances of your case. Rather, this general information is being provided as a courtesy to you for informational and educational purposes only. It is no substitute for the advice of an attorney.
You have to understand that supervised visitation by its very nature was intended to be temporary. Indeed it only lasts as long as there is a real danger present. It's temporary because it is so restrictive and because it creates a sense of fear or hesitancy in the child when there has to be a third person present. On some level a child could begin to think "what's wrong with me" and "why do I need somebody else here with my own parent?" The three examples you describe are not a typical of what many children do when they're young. And the fact that the father has not had much experience or hands-on time with the child may explain why he has not used the best judgment at all times. No parent is perfect all the time. The older of a child gets the less they will need supervised visitation. No one would expect the court to order supervised visitation at the child was 16 or 17 years old. So the question is not whether the court will continue to order supervised visitation but how long before the court allows unsupervised visitation. And if that's the case you may wish to start thinking about how you can help the father be a better parent before you put yourself in a situation where the chalice going to simply have the opportunity to find out for himself what kind of a dad he was given at birth.